Our Trysts with `Bal' and Noncommutative Geometry

TL;DR

This paper reflects on Prof. A.P. Balachandran's influence and presents a recent formulation of quantum mechanics on (1+1)D noncommutative space-time using Hilbert-Schmidt operators, illustrating geometrical phase emergence.

Contribution

It introduces a new formulation of quantum mechanics on noncommutative space-time employing Hilbert-Schmidt operators and explores geometrical phases in this context.

Findings

Demonstration of geometrical phase in a time-dependent harmonic oscillator

Formulation of quantum mechanics on (1+1)D noncommutative space-time

Application of Hilbert-Schmidt operators in noncommutative quantum mechanics

Abstract

This contributory article begins with our fond and sincere reminiscences about our beloved Prof. A.P. Balachandran. In the main part, we discuss our recent formulation of quantum mechanics on (1+1)D noncommutative space-time using Hilbert-Schmidt operators. As an application, we demonstrate how geometrical phase in a system of time-dependent forced harmonic oscillator living in the Moyal space-time can emerge.